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We analyze data from multiple flybys by the Solar Orbiter, BepiColombo, and Parker Solar Probe (PSP) missions to study the interaction between Venus' plasma environment and the solar wind forming the induced magnetosphere. Through examination of magnetic field and plasma density signatures we characterize the spatial extent and dynamics of Venus' magnetotail, focusing mainly on boundary crossings. Notably, we observe significant differences in boundary crossing location and appearance between flybys, highlighting the dynamic nature of Venus' magnetotail. In particular, during Solar Orbiter's third flyby, extreme solar wind conditions led to significant variations in the magnetosheath plasma density and magnetic field properties, but the increased dynamic pressure did not compress the magnetotail. Instead, it is possible that the increased EUV flux at this time rather caused it to expand in size. Key findings also include the identification of several far downstream bow shock (BS), or bow wave, crossings to at least 60 Rv (1 Rv = 6,052 km is the radius of Venus), and the induced magnetospheric boundary to at least 20 Rv. These crossings provide insight into the extent of the induced magnetosphere. Pre-existing models from Venus Express were only constrained to within ~5 Rv of the planet, and we provide modifications to better fit the far-downstream crossings. The new model BS is now significantly closer to the central tail than previously suggested, by about 10 Rv at 60 Rv downstream.

A LAnKe (also known as a Lie algebra of the $n$th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $\mathfrak{S}_m$ acts on the multilinear part of the free LAnKe on $m=(n-1)k+1$ generators, where $k$ is the number of brackets. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for $n \ge k$, its irreducible decomposition contains no summand whose Young diagram has at most $k-1$ columns. It is known that the answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$.

A LAnKe (also known as a Filippov algebra or a Lie algebra of the $n$-th kind) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear component of the free LAnKe on $2n-1$ generators as an irreducible representation. They have claimed that the multilinear component on $3n-2$ generators decomposes as a direct sum of two irreducible symmetric group representations. This paper provides a proof of the later statement., Comment: Mainly typos corrected and notation streamlined

New presentations of Specht modules of symmetric groups over fields of characteristic zero have been obtained by Brauner, Friedmann, Hanlon, Stanley and Wachs. These involve generators that are column tabloids and relations that are Garnir relations with maximal number of exchanges between consecutive columns or symmetrization of Garnir relations with minimal number of exchanges between consecutive columns. In this paper, we examine Garnir relations and their symmetrization with any number of exchanges. In both cases, we provide sufficient arithmetic conditions so that the corresponding quotient is a Specht module. In particular, in the first case this yields new presentations of Specht modules if the parts of the conjugate partition that correspond to maximal number of exchanges greater than 1 are distinct. These results generalize the presentations mentioned above and offer an answer to a question of Friedmann, Hanlon and Wachs. Our approach is via representations of the general linear group., Comment: This version of the paper differs from the published version as follows. A typo in the statement of Corollary 4.4 and Corollary 6.1 has been corrected, instead of $j=1,\dots, b$, it should be $j=1, \dots, k$. Also, in the statement of Theorem 6.2 it should be $j=1,\dots, \mu_{c+1}$. Examples 6.3(1) and 6.3(3) have been corrected accordingly

In this paper, we study finiteness criteria for the Gorenstein homological dimension of groups over a commutative ring of finite weak Gorenstein global dimension and provide good estimations for the Gorenstein weak global dimension of group rings. As a result, we obtain Gorenstein analogues of well known properties in classical homological algebra over some large families of infinite groups. Moreover, we prove that over a commutative ring of finite weak Gorenstein global dimension, the Gorenstein cohomological dimension of a group $G$ bounds the Gorenstein homological dimension of $G$. Finally, we compare the generalized cohomological dimension and the generalized homological dimension of a group, defined by Ikenaga., Comment: 20 pages. arXiv admin note: text overlap with arXiv:2302.06022

Consider the general linear group $G=GL_{n}(K)$ defined over an infinite field $K$ of positive characteristic $p$. We denote by $\Delta(\lambda)$ the Weyl module of $G$ which corresponds to a partition $\lambda$. Let $\lambda, \mu $ be partitions of $r$ and let $\gamma$ be partition with all parts divisible by $p$. In the first main result of this paper, we find sufficient conditions on $\lambda, \mu$ and $\gamma$ so that $Hom_G(\Delta(\lambda),\Delta(\mu))$ $ \simeq$ $ Hom_G(\Delta(\lambda +\gamma),\Delta(\mu +\gamma))$, thus providing an answer to a question of D. Hemmer. As corollaries we obtain stability and periodicity results for homomorphism spaces. In the second main result we find related sufficient conditions on $\lambda, \mu$ and $p$ so that $Hom_G(\Delta(\lambda),\Delta(\mu))$ is nonzero. An explicit map is provided that corresponds to the sum of all semistandard tableaux of shape $\mu$ and weight $\lambda$., Comment: The referee's comments and suggestions have been incorporated. Accepted for publication in Algebraic Combinatorics

We give characterizations of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over the group algebra for large families of infinite groups and show that every weak Gorenstein projective, weak Gorenstein flat and weak Gorenstein injective module is Gorenstein projective, Gorenstein flat and Gorenstein injective, respectively. These characterizations provide Gorenstein analogues of Benson's cofibrant modules. We deduce that, over a commutative ring of finite Gorenstein weak global dimension, every Gorenstein projective module is Gorenstein flat. Moreover, we study cases where the tensor product and the group of homomorphisms between modules over the group algebra is a Gorenstein module. Finally, we determine the Gorenstein homological dimension of an $\textsc{\textbf{lh}}\mathfrak{F}$-group over a commutative ring of finite Gorenstein weak global dimension., Comment: 21 pages

Projectively coresolved Gorenstein flat modules were introduced recently by Saroch and Stovicek and were shown to be Gorenstein projective. While the relation between Gorenstein projective and Gorenstein flat modules is not well understood, the class of projectively coresolved Gorenstein flat modules is contained in the class of Gorenstein flat modules. This paper proves necessary and sufficient conditions for a module of finite Gorenstein flat dimension to be projectively coresolved Gorenstein flat, or of finite flat dimension. Stability results for the class of projectively coresolved Gorenstein flat modules are also established., Comment: 17 pages

The spatial resolution of images of living samples obtained by fluorescence microscopes is physically limited due to the diffraction of visible light, which makes the study of entities of size less than the diffraction barrier (around 200 nm in the x-y plane) very challenging. To overcome this limitation, several deconvolution and super-resolution techniques have been proposed. Within the framework of inverse problems, modern approaches in fluorescence microscopy reconstruct a super-resolved image from a temporal stack of frames by carefully designing suitable hand-crafted sparsity-promoting regularisers. Numerically, such approaches are solved by proximal gradient-based iterative schemes. Aiming at obtaining a reconstruction more adapted to sample geometries (e.g. thin filaments), we adopt a plug-and-play denoising approach with convergence guarantees and replace the proximity operator associated with the explicit image regulariser with an image denoiser (i.e. a pre-trained network) which, upon appropriate training, mimics the action of an implicit prior. To account for the independence of the fluctuations between molecules, the model relies on second-order statistics. The denoiser is then trained on covariance images coming from data representing sequences of fluctuating fluorescent molecules with filament structure. The method is evaluated on both simulated and real fluorescence microscopy images, showing its ability to correctly reconstruct filament structures with high values of peak signal-to-noise ratio (PSNR).

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group $G$ over a commutative ring $R$ and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of $RG$ in terms of this dimension and the Gorenstein global dimension of $R$. Moreover, we study special cases of groups, such as $\textsc{\textbf{lh}}\mathfrak{F}$-groups, and show that for such a group every Gorenstein projective $RG$-module is Gorenstein flat when the global dimension of $R$ is finite., Comment: 32 pages

In this paper we study periodicity phenomena for modular extensions between Weyl modules and between Weyl and simple modules of the general linear group that are associated to adding a power of the characteristic to the first parts of the involved partitions., Comment: Referee's comments incorporated. To appear in Mathematische Zeitschrift

The stability of the class of projectively coresolved Gorenstein flat modules, under the very Gorenstein process used to define them, is proven in this paper. Moreover, a new characterization of the projectively coresolved Gorenstein flat dimension is given., Comment: Unpublished manuscript

We propose a 3D super-resolution approach to improve both lateral and axial spatial resolution in Total Internal Reflectance Fluorescence (TIRF) imaging applications. Our approach, called 3D-COL0RME (3D - Covariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation) improves both lateral and axial resolution by combining sparsity-based modelling for precise molecule localisation and intensity estimation in the lateral plane with a 3D reconstruction procedure in the axial one using Multi-Angle TIRF (MA-TIRF). Differently from state-of-the-art approaches, 3D-COL0RME does not require the use of special equipment as it can be used with standard fluorophores. We validate 3D-COL0RME on simulated MA-TIRF blinking-type data and on challenging real MA-TIRF acquisitions, showing significant resolution improvements., Comment: Published in 2022 IEEE 19th International Symposium on Biomedical Imaging (ISBI)

To overcome the physical barriers caused by light diffraction, super-resolution techniques are often applied in fluorescence microscopy. State-of-the-art approaches require specific and often demanding acquisition conditions to achieve adequate levels of both spatial and temporal resolution. Analyzing the stochastic fluctuations of the fluorescent molecules provides a solution to the aforementioned limitations, as sufficiently high spatio-temporal resolution for live-cell imaging can be achieved by using common microscopes and conventional fluorescent dyes. Based on this idea, we present COL0RME, a method for COvariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation, which achieves good spatio-temporal resolution by solving a sparse optimization problem in the covariance domain and discuss automatic parameter selection strategies. The method is composed of two steps: the former where both the emitters' independence and the sparse distribution of the fluorescent molecules are exploited to provide an accurate localization; the latter where real intensity values are estimated given the computed support. The paper is furnished with several numerical results both on synthetic and real fluorescence microscopy images and several comparisons with state-of-the art approaches are provided. Our results show that COL0RME outperforms competing methods exploiting analogously temporal fluctuations; in particular, it achieves better localization, reduces background artifacts and avoids fine parameter tuning., Comment: Accepted manuscript in Biological Imaging and Published online by Cambridge University Press on 16 February 2022. 29 pages and 17 figures

Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions of $n$, where $\lambda=(\lambda_1,...,\lambda_n)$ and $\mu=(\mu_1,..,\mu_n)$. By $S^{\lambda}$ we denote the Specht module corresponding to $\lambda$ for the group algebra $K\mathfrak{S}_n$ of the symmetric group $\mathfrak{S}_n$. D. Hemmer has raised the question of relating the homomorphism spaces $\Hom_{\mathfrak{S}_n}(S^{\mu}, S^{\lambda})$ and $\Hom_{\mathfrak{S}_{n'}}(S^{\mu^+}, S^{\lambda^+})$, where $n'=n+kp^d$, $\lambda^+ =\lambda+(kp^{d})$, $\mu^+=\mu+(kp^{d})$, and $d, k$ are positive integers. We show that these are isomorphic if $p$ is odd, $p^d >\min\{\lambda_2, \mu_1-\lambda_1\}$ and $\mu_2 \le \lambda_1$., Comment: This paper is superseded by arXiv:2211.06978

Let $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$. Also for each $p$ we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2., Comment: Typos corrected. Minor changes in Section 6 for clarity

Super-resolution light microscopy overcomes the physical barriers due to light diffraction, allowing for the observation of otherwise indistinguishable subcellular entities. However, the specific acquisition conditions required by state-of-the-art super-resolution methods to achieve adequate spatio-temporal resolution are often very challenging. Exploiting molecules fluctuations allows good spatio-temporal resolution live-cell imaging by means of common microscopes and conventional fluorescent dyes. In this work, we present the method COL0RME for COvariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation. It codifies the assumption of sparse distribution of the fluorescent molecules as well as the temporal and spatial independence between emitters via a non-convex optimization problem formulated in the covariance domain. In order to deal with real data, the proposed approach also estimates background and noise statistics. It also includes a final estimation step where intensity information is retrieved, which is valuable for biological interpretation and future applications to super-resolution imaging.

Let $k$ be an infinite field of positive characteristic. We determine all homomorphisms between Weyl modules for $GLn(k)$, where one of the partitions is a hook. As a consequence we obtain a nonvanishing result concerning homomorphisms between Weyl modules for algebraic groups of type B, C and D when one of the partitions is a hook., Comment: Updated with reviewers' suggestions. Subsection 3.2 was added. Minor corrections. To appear in J. Algebra

This paper concerns representations of the integral general linear group. The extension groups $Ext^2$ between any pair of hook Weyl modules are determined via a detailed study of cyclic generators and relations associated to certain extensions. As a corollary, the modular extension groups $Ext^1$ between such modules are determined., Comment: 22 pages

We determine the integral extension groups $Ext^1({\Delta}(h),{\Delta}(h(k)))$ and $Ext^k({\Delta}(h),{\Delta}(h(k)))$, where ${\Delta}(h),{\Delta}(h(k))$ are the Weyl modules of the general linear group $GL_n$ corresponding to the hook partitions $h=(a,1^b)$, $h(k)=(a+k,1^{b-k})$., Comment: The paper has been updated with the reviewer's suggestions and comments. In particular the Remark after Theorem 4.1 has been corrected. To appear in Journal of Pure and Applied Algebra

The objective of this White Paper submitted to ESA's Voyage 2050 call is to get a more holistic knowledge of the dynamics of the Martian plasma system from its surface up to the undisturbed solar wind outside of the induced magnetosphere. This can only be achieved with coordinated multi-point observations with high temporal resolution as they have the scientific potential to track the whole dynamics of the system (from small to large scales), and they constitute the next generation of Mars' exploration as it happened at Earth few decades ago. This White Paper discusses the key science questions that are still open at Mars and how they could be addressed with coordinated multipoint missions. The main science questions are: (i) How does solar wind driving impact on magnetospheric and ionospheric dynamics? (ii) What is the structure and nature of the tail of Mars' magnetosphere at all scales? (iii) How does the lower atmosphere couple to the upper atmosphere? (iv) Why should we have a permanent in-situ Space Weather monitor at Mars? Each science question is devoted to a specific plasma region, and includes several specific scientific objectives to study in the coming decades. In addition, two mission concepts are also proposed based on coordinated multi-point science from a constellation of orbiting and ground-based platforms, which focus on understanding and solving the current science gaps., Comment: White Paper submitted to ESA's Voyage 2050 Call, 27 pages, 8 Figures, 2 Tables